----------------------------------------------------- # # # Chapter 4: Brownian Motion, Wiener Measure # # and the Black-Scholes Model # # # ----------------------------------------------------- Summary: The returns of liquidly tradable assets like stocks, stock indices (or futures on them), currencies and commodities like crude oil, gold and silver have some typical properties which are common to all of them. If the returns are normalized, that is, if we subtract the mean and divide by the standard deviation, their distribution is close to a normal distribution. This leads to S(t_k) = S(t_{k-1}) * (1 + mean + stddev * phi_k ) (1) with some standard normal distributed random numbers phi_k and S(t_k) denoting the closing price on day t_k. If the returns are not considered on a daily basis, but over some time horizon Delta_t, a statistical analysis reveals that the mean is actually proportional to Delta_t and the standard deviation stddev is propor- tional to sqr(Delta_t). Thus one is lead to S(t_k) = S(t_{k-1}) * (1 + mean * Delta_t + stddev * sqr(Delta_t) * phi_k ) (2) with t_k = k*Delta_t. Equation (2) is the time discrete version of the Black Scholes model. The corresponding product measure of Gaussian densities for the random numbers phi_k lead in a natural way to the concept of Brownian motion and Wiener measure. The geometric Brownian motion arises as the explicit (and, in discrete time, approximate) solution of the stochastic difference equation (2). The term "geometric Brownian motion" and "Black-Scholes model" actually refer to the same stochastic process. A data analysis with several plots and histograms for the returns of SP500, DAX30 and the GE-stock can be found here:http://time-series.de/stylized-facts-financial-time-series.html

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